Applications of Fourier analysis to convex geometry

Aleksandr Koldobsky的这项数学研究项目涉及到基于凸体的截面和投影信息来研究凸体的几何性质。这一研究领域在数学和科学的许多领域都有重要的应用。基于傅立叶分析方法，Koldobsky最近发展了一种新的方法来研究凸体的截面和投影。其思想是用傅里叶变换来表示物体的某些几何特征，然后用调和分析的方法来解决几何问题。这种方法已经得到了几个结果，包括Busemann-Petty和Shephard问题的傅立叶解析解，交点和投影体的特征。Koldobsky将把傅里叶方法应用于该领域的其他公开问题，重点是在体积比较问题中新发现的稳定性和分离性，以及它们在超平面猜想中的应用。另一个方向是研究截面体和投影体之间的对偶性，特别是建立交投影体和极投影体的同构等价。概率论中的一个老问题是刻画所有随机向量的性质，即坐标的所有线性组合都具有相同的分布，直到一个常数。经典的例子是稳定的随机向量。虽然亚历山大·柯尔多布斯基在这个数学研究项目中考虑的问题涉及三个不同的数学领域(凸几何、泛函分析和概率)，但大多数结果的求解策略是共同的--这些问题被转化为傅立叶变换的语言，然后作为调和分析的问题来处理。在凸几何中，Koldobsky考虑了使用关于实体的平面截面和投影的数据来重建实体的问题。这些结果和方法将经典的X射线层析成像技术扩展到只有特殊的切片或投影集合可用的情况。在《概率论》一书中，Koldobsky研究了稳定过程，这是继承了正态规律的主要自我再生产性质的概率规律。Koldobsky开发的基本傅立叶技术已经导致了对其他学科的重要应用(例如，图像处理中的神经网络；例如，人们可以仅使用关于信号在相同长度的时间间隔上的最大幅度的信息来找到该信号的主要频率)。该项目的一个重要部分是研究生和博士后的参与和培训。

This mathematics research project by Aleksandr Koldobsky concerns the study of geometric properties of convex bodies based on information about sections and projections of these bodies. This research area has important applications to many areas of mathematics and science. Based on methods of Fourier analysis, Koldobsky has recently developed a new approach to the study of sections and projections of convex bodies. The idea is to express certain geometric characteristics of a body in terms of the Fourier transform and then use methods of harmonic analysis to solve geometric problems. This approach has led to several results including Fourier analytic solutions of the Busemann-Petty and Shephard problems,characterizations of intersection and projection bodies. Koldobsky will apply the Fourier approach to other open problems in the area, with the emphasis on newly discovered stability and separation properties in volume comparison problems and their applications to the hyperplane conjecture. Another direction is to study the duality between section and projections, in particular to establish isomorphic equivalence of intersection and polar projection bodies. An old problem in probability theory is to characterize all random vectors having the property that all linear combinations of coordinates have the same distribution, up to a constant. The classical examples are stable random vectors. Koldobsky will characterize all random vectors with this property.While the problems considered in this mathematics research project by Aleksandr Koldobsky are related to three different areas of mathematics (convex geometry, functional analysis and probability) the strategy of solution is common for most of the results - the questions are translated into the language of the Fourier transform and then treated as problems in harmonic analysis. In convex geometry, Koldobsky considers the problem of reconstructing a solid using data about plane sections and projections of this solid. The results and methods extend the classical techniques of x-ray tomography to situations where only special sets of sections or projections are available. In probability, Koldobsky studies stable processes, which are probabilistic laws that inherit the main self-reproductive property of the normal law. The underlying Fourier techniques developed by Koldobsky have led to important applications to other disciplines (e.g., neural networks in image processing; for example, one can find the main frequencies of a signal using only information about the maximal amplitude of this signal on intervals of time of equal lengths). An important part of the project is the involvement and training of graduate students and postdocs.